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    The discovery of consistent non-Euclidean geometries (Rie... — Carmelics
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    Challenges→Euclidean geometry possesses certainty and necessity

    The discovery of consistent non-Euclidean geometries (Riemann, Lobachevsky) demonstrates that Euclid's parallel postulate is not logically necessary.

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    Reasons For

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    • 1.If a mathematical system is internally consistent without the parallel postulate, that postulate cannot be logically necessary for geometric truth.
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    • 2.Euclid presented the parallel postulate separately from other axioms, suggesting even he recognized it as less self-evident than logical necessities.
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    • 3.Non-Euclidean geometries accurately model physical space in certain contexts (curved spacetime), proving necessity depends on the domain, not pure logic.
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    Reasons Against

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    • 1.Logical necessity and empirical consistency are distinct: non-Euclidean systems may be consistent without showing Euclid's postulate isn't necessary for Euclidean space itself.
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    • 2.The parallel postulate may be logically necessary specifically for flat plane geometry, even if other geometries are possible—necessity is domain-relative, not universal.
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    Related

    Euclid presented the parallel postulate separately from other axioms, suggesting...Euclidean geometry possesses certainty and necessityIf a mathematical system is internally consistent without the parallel postulate...Logical necessity and empirical consistency are distinct: non-Euclidean systems ...
    +2 moreShow less
    Non-Euclidean geometries accurately model physical space in certain contexts (cu...The parallel postulate may be logically necessary specifically for flat plane ge...

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